symmetry in crystals
DESCRIPTION
Symmetry in crystals. Infinitely repeating lattices. An integral number of unit translations along any axis will arrive at an identical point. A unit translation along any axis will arrive at an identical point. The composition of each unit should be identical. - PowerPoint PPT PresentationTRANSCRIPT
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Symmetry in crystals
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Infinitely repeating lattices
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An integral number of unit translations
along any axis will arrive at
an identical point.
![Page 5: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/5.jpg)
A unit translation along any axis will
arrive at an identical point
![Page 6: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/6.jpg)
The composition of each unit should
be identical.
![Page 7: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/7.jpg)
A unit translation parallel to any axis will
arrive at an identical point
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Face centered lattice
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Unit Cell
![Page 13: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/13.jpg)
3 axes, a, b, c and 3 angles , , and
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4 3-fold axes along diagonals
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4-fold axis
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3 2-fold axes
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1 6-fold axis
6-fold
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A cube with 1 diagonal shortened or lengthened.
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3-fold axis
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1 2-fold axis
2-fold
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2-fold axis
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2-fold axis3-fold axis4-fold axis6-fold axis
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Symmetry in Crystals
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Rotational symmetry
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Rotational symmetry
Possible: 2, 3, 4, 6 - fold axes
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Rotational inversion
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Mirror plane
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Screw axes:
a combination
of rotation and
translation.
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Screw axes:
a combination
of rotation and
translation.
21 screw = 180o rotation + 1/2 cell translation
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31 screw =
120o rotation +
1/3 unit translation
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Glide plane:
a combination
of mirror and
translational
symmetry.
![Page 46: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/46.jpg)
Glide plane:
a combination
of mirror and
translational
symmetry.
1/2 unit translation
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Given the 7 crystal systems
and various symmetry operations,
the number of ways a continuously
repeating lattice can be formed is
limited.
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Theoretical studies of the
geometries of crystals
completed in 1890 demonstrated
that there are 230 ways to put
together an infinitely repeating
lattice.
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Unit Cell
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Unit Cell
Space group P1
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Unit Cell
P1 = primitive cell + inversion center
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Unit Cell
P1 = primitive cell + inversion center
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Unit Cell
x, y, z = 0, 0, 0
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Unit Cell
x, y, z = 1, 0, 0
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Unit Cell
x, y, z = 0, 1, 0
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Unit Cell
x, y, z = 0, 0, 1
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Unit Cell
x, y, z = 1, 0, 1
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Unit Cell
x, y, z = 1, 1, 1
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Unit Cell
P1 = primitive cell + inversion center
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Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0
-x, -y, -z -0, -0, -0
Fractional coordinates: the fraction onemust move along each axis to arrive at a point.
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Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0 1, 0, 0
-x, -y, -z -0, -0, -0 -1, -0, -0
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Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0 1, 0, 0
-x, -y, -z -0, -0, -0 -1, -0, -0
An integral number of unit translations results in an identical point in the lattice.
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Unit Cell
P1 = primitive cell + inversion center
1/2, 1/2, 1/2
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Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0 1/2, 1/2, 1/2
-x, -y, -z -0, -0, -0 -1/2, -1/2, -1/2
An integral number of unit translations results in an identical point in the lattice.
![Page 65: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/65.jpg)
Unit Cell
P1 = primitive cell + inversion center
1, 1, 1/2
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Inversion Center
Cartesian Coordinates:
x, y, z 0, 0, 0 1, 1, 1/2
-x, -y, -z -0, -0, -0 -1, -1, -1/2
An integral number of unit translations results in an identical point in the lattice.
![Page 67: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/67.jpg)
What causes crystals to
form and take a particular structure?
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Strong Forces:
Electrostatic forces in ionic crystals.
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NaCl
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NaCl+ -
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NaCl+ -
+ +
+ +
++
- -
--
- -
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NaCl ionic bond energy
is 785 kj/mol.
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NaCl CsCl
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NaCl CsCl
Na+ 1.00 Å Cl- 1.81 Å Cs+ 1.69 Å
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NaCl CsCl
Two different cells; same charges;
same stoichiometry.
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Determining the contents of
the unit cell.
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NaCl
Ion within cell = 1 per cell
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NaCl
Ion on face of cell = 1/2 per cell
Ion within cell = 1 per cell
(shared with 2 cells)
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NaCl
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion within cell = 1 per cell
(shared with 4 cells)
![Page 82: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/82.jpg)
NaCl
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
Ion within cell = 1 per cell
(shared with 8 cells)
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NaCl
Ion on face of cell = 1/2 per cell 0 6
Ion on edge of cell = 1/4 per cell 12 0
Ion at corner of cell = 1/8 per cell 0 8
Na+ Cl-
1 0Ion within cell = 1 per cell
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NaCl
Ion on face of cell = 1/2 per cell 0 6
Ion on edge of cell = 1/4 per cell 12 0
Ion at corner of cell = 1/8 per cell 0 8
Na+ Cl-
1 0
Total ions in cell: Na+ Cl-
1 3
Ion within cell = 1 per cell
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NaCl
Ion on face of cell = 1/2 per cell 0 6
Ion on edge of cell = 1/4 per cell 12 0
Ion at corner of cell = 1/8 per cell 0 8
Na+ Cl-
1 0
Total ions in cell: Na+ Cl-
1 3 3 1
Ion within cell = 1 per cell
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NaCl
Ion on face of cell = 1/2 per cell 0 6
Ion on edge of cell = 1/4 per cell 12 0
Ion at corner of cell = 1/8 per cell 0 8
Na+ Cl-
1 0
Total ions in cell: Na+ Cl-
1 3 3 1
Z = 4
Ion within cell = 1 per cell
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Determining ionic radii using
crystal structures.
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CsCl
![Page 91: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/91.jpg)
CsCl
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
Ion within cell = 1 per cell
![Page 92: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/92.jpg)
CsCl
Ion on face of cell = 1/2 per cell 0 0
Ion on edge of cell = 1/4 per cell 0 0
Ion at corner of cell = 1/8 per cell 0 8
1 0
Cs+ Cl-
Ion within cell = 1 per cell
![Page 93: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/93.jpg)
CsCl
Ion on face of cell = 1/2 per cell 0 0
Ion on edge of cell = 1/4 per cell 0 0
Ion at corner of cell = 1/8 per cell 0 8
1 0
Cs+ Cl-
Z = 1
Ion within cell = 1 per cell
![Page 94: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/94.jpg)
Ionic crystals are held together
by strong electrostatic forces.
The crystal unit cell is influenced
by ionic sizes.
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CaCl2
![Page 96: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/96.jpg)
Ion within cell = 1 per cell
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
CaCl2
![Page 97: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/97.jpg)
Ion within cell = 1 per cell
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
CaCl2
![Page 98: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/98.jpg)
Ion within cell = 1 per cell
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
CaCl2
![Page 99: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/99.jpg)
Ion within cell = 1 per cell
Ion on face of cell = 1/2 per cell
Ion on edge of cell = 1/4 per cell
Ion at corner of cell = 1/8 per cell
CaCl2
![Page 100: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/100.jpg)
Ion within cell = 1 per cell 1 2
Ion on face of cell = 1/2 per cell 0 4
Ion on edge of cell = 1/4 per cell 0 0
Ion at corner of cell = 1/8 per cell 8 0
CaCl2
Ca2+ Cl-
![Page 101: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/101.jpg)
Ion within cell = 1 per cell 1 2
Ion on face of cell = 1/2 per cell 0 4
Ion on edge of cell = 1/4 per cell 0 0
Ion at corner of cell = 1/8 per cell 8 0
CaCl2
Ca2+ Cl-
Z = 2
![Page 102: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/102.jpg)
Diamond
![Page 103: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/103.jpg)
The strengths of chemical bonds: kJ/mol Weak < 200
Average <500>
Strong >800
![Page 104: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/104.jpg)
Diamond
C - C covalent bond = 1.544 Å
Bond enthalpy 348 kJ/mol
![Page 105: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/105.jpg)
Diamond
C - C covalent bond = 1.544 Å
Bond enthalpy 348 kJ/mol
![Page 106: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/106.jpg)
Diamond
Atom within cell = 1 per cell
Atom on face of cell = 1/2 per cell
Atom on edge of cell = 1/4 per cell
Atom at corner of cell = 1/8 per cell
![Page 107: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/107.jpg)
Diamond
Atom within cell = 1 per cell 4
Atom on face of cell = 1/2 per cell
Atom on edge of cell = 1/4 per cell
Atom at corner of cell = 1/8 per cell
C
![Page 108: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/108.jpg)
Diamond
Atom within cell = 1 per cell 4
Atom on face of cell = 1/2 per cell 6
Atom on edge of cell = 1/4 per cell
Atom at corner of cell = 1/8 per cell
C
![Page 109: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/109.jpg)
Diamond
Atom within cell = 1 per cell 4
Atom on face of cell = 1/2 per cell 6
Atom on edge of cell = 1/4 per cell 0
Atom at corner of cell = 1/8 per cell
C
![Page 110: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/110.jpg)
Diamond
Atom within cell = 1 per cell 4
Atom on face of cell = 1/2 per cell 6
Atom on edge of cell = 1/4 per cell 0
Atom at corner of cell = 1/8 per cell 8
C
![Page 111: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/111.jpg)
Diamond
Atom within cell = 1 per cell 4
Atom on face of cell = 1/2 per cell 6
Atom on edge of cell = 1/4 per cell 0
Atom at corner of cell = 1/8 per cell 8
C
Z = 8
![Page 112: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/112.jpg)
Molecular Crystals
![Page 113: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/113.jpg)
Molecular Crystals:
Consist of repeating arrays
of molecules and/or ions.
![Page 114: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/114.jpg)
![Page 115: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/115.jpg)
![Page 116: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/116.jpg)
C17H24NO2+ Cl- . 3 H2O
![Page 117: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/117.jpg)
V = 974.45 Å3
C17H24NO2+ Cl- . 3 H2O FW = 363.87 g/mol
Z = 2
Density =727.74 g
5866.19 x 10-1cm3
Density =363.87 g (2)
974.45 x 10-24 x 6.02 x 1023
= 1.241 g/cm3
![Page 118: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/118.jpg)
C17H24NO2+ Cl- . 3 H2O
![Page 119: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/119.jpg)
![Page 120: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/120.jpg)
Although Z = 2, the unit cell containsportions of a number of molecules.
![Page 121: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/121.jpg)
Cl-
![Page 122: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/122.jpg)
Cl-
![Page 123: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/123.jpg)
Cl-
H2O
![Page 124: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/124.jpg)
Cl-
H2O
Hydrogen bondsCl OH2
![Page 125: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/125.jpg)
![Page 126: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/126.jpg)
Hydrogen bond
![Page 127: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/127.jpg)
![Page 128: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/128.jpg)
Model with atoms having VDW radii.
![Page 129: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/129.jpg)
![Page 130: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/130.jpg)
C17H24NO2+ Cl- . 3 H2O
Although this material is ionic, the + and - chargesare not close enough tocontribute to the formationof the crystal.
![Page 131: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/131.jpg)
Molecular crystals tend to be
held together by forces weaker than
chemical bonds.
van der Waal’s forces are always
a factor.
Hydrogen bonding is often present.
![Page 132: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/132.jpg)
![Page 133: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/133.jpg)
A layer in an ionic solid with ionsof similar radii.
![Page 134: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/134.jpg)
![Page 135: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/135.jpg)
![Page 136: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/136.jpg)
![Page 137: Symmetry in crystals](https://reader035.vdocument.in/reader035/viewer/2022081420/56813485550346895d9b6701/html5/thumbnails/137.jpg)